Analog to digital convertion

To explain the analog-to-digital conversion process, we will assume that the input signal is a cosine wave with frequency \(F\), angular frequency \(\Omega\) and amplitude \(a\).

\[x\left(t\right) = a \cos\left(\Omega t + \phi\right) = a \cos\left(2\pi F t + \phi\right)\]

Obtaining

\[x\left[n\right] = a \cos\left(\omega n + \phi\right) = a \cos\left(2\pi f n + \phi\right)\]

Analog to digital convertion

Analog to digital convertion

Replacing in previous equations, we have the expression:

\[x[n] = x(nT_s) = a \cos\left( 2\pi F n T_s + \phi \right) = a \cos\left( 2\pi n \frac{F}{F_s} + \phi \right) \]

Where:

\[\omega = \Omega T_s, \quad f = \frac{F}{F_s}\]

Sample and quantization of an ECG signal

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import chirp

# Generate a synthetic ECG-like signal (chirp function as approximation)
fs_original = 10000  # High sampling rate (Hz) - "continuous" signal
t = np.linspace(0, 1, fs_original, endpoint=False)  # 1-second signal
signal = np.sin(2 * np.pi * 1.7 * (t**2))  # Simulated chirp (similar to ECG waves)

# Downsample (Sampling Process)
fs_sampled = 200  # Sampling frequency in Hz (e.g., ECG sampled at 200 Hz)
t_sampled = np.arange(0, 1, 1/fs_sampled)
signal_sampled = np.sin(2 * np.pi * 1.7 * (t_sampled**2))

# Quantization (8-bit and 4-bit)
def quantize(signal, bits):
    levels = 2**bits
    min_val, max_val = signal.min(), signal.max()
    step = (max_val - min_val) / levels
    quantized_signal = np.round((signal - min_val) / step) * step + min_val
    return quantized_signal

signal_quantized_8bit = quantize(signal_sampled, 8)
signal_quantized_4bit = quantize(signal_sampled, 4)

# Plot Results
plt.figure(figsize=(12, 6))

# Original vs Sampled Signal
plt.subplot(2, 1, 1)
plt.plot(t, signal, 'k', alpha=0.3, label='Original Signal (High Resolution)')
plt.plot(t_sampled, signal_sampled, 'ro-', label=f'Sampled Signal ({fs_sampled} Hz)')
plt.xlabel("Time (s)")
plt.ylabel("Amplitude")
plt.legend()
plt.title("Sampling Process")

# Quantized Signals
plt.subplot(2, 1, 2)
plt.plot(t_sampled, signal_sampled, 'bo-', alpha=0.5, label="Original Sampled")
plt.plot(t_sampled, signal_quantized_8bit, 'go-', label="Quantized 8-bit")
plt.plot(t_sampled, signal_quantized_4bit, 'ro-', label="Quantized 4-bit")
plt.xlabel("Time (s)")
plt.ylabel("Amplitude")
plt.legend()
plt.title("Quantization Effect")

plt.tight_layout()
plt.show()

Slide 1 — Course framing

Practical Sampling & Quantization for Biomedical Signals

• Focus: sampling, aliasing, reconstruction, quantization, SNR, range, companding
• Clinical touchstones: ECG, EEG, EMG, PPG
• Constraints: ≤7 lines/slide, Python plots only, no Fourier content
• Primary source: Oppenheim, Willsky, Nawab, Signals and Systems, 2nd ed. (ISBN: 978-0138147570)
• Secondary: Shannon (DOI: 10.1109/JRPROC.1949.232969), Nyquist (DOI: 10.1109/T-AIEE.1928.5055024), Jayant & Noll (ISBN: 0-13-211913-7)

Slide 2 — Sampling model & Nyquist (with ECG example)

::: columns ::: {.column width=45%} Model

• Sample period: $T_s=1/f_s$ (Oppenheim 2e, ISBN: 978-0138147570)
• Impulse train view: $x_s(t)=_{n=-}^{}x(nT_s),(t-nT_s)$ (Oppenheim 2e)
• Bandlimit $B$ ⇒ Nyquist: $f_s2B$ (Shannon 1949, DOI above) ::: ::: {.column width=45%} Clinical tie-in (ECG)
• Diagnostic ECG bandwidth $B $ ⇒ pick $f_s=500 $ for margin
• Rationale: $f_s2B$ + antialias margin improves filter realizability (Shannon 1949; Webster MI 4e, ISBN: 978-0471676003) ::: :::

Slide 3 — Aliasing & anti-alias filtering (with EMG example)

::: columns ::: {.column width=45%} Aliasing

• Frequencies above $f_s/2$ fold into baseband as false low-frequency content (Oppenheim 2e)
• Prevent by pre-sampling low-pass: $f_c< f_s/2$ with transition guard band ::: ::: {.column width=45%} Clinical tie-in (EMG)
• Surface EMG has content up to $5001000 $
• Example: choose $f_s=2 $, $f_c $ to limit folding (Webster MI 4e, ISBN above) ::: :::

Slide 4 — Discrete-time reconstruction (conceptual)

• Goal: recover a continuous-time $y(t)$ consistent with samples $x[n]$
• Ideal: band-limited interpolation + smoothing LPF (conceptual; no Fourier used)
• Practical DAC: zero-order hold followed by analog smoothing LPF (Oppenheim 2e)
• Clinical tie-in (PPG): reconstruct smooth pulsatile waveform for heart-rate variability after downlink

Slide 5 — Uniform quantization & step size $$ (with ECG example)

::: columns ::: {.column width=45%} Definitions

• ADC input range $[V_{},V_{}]$, bit depth $b$
• Step size: \(\Delta=\frac{V\_{\max}-V\_{\min}}{2^{b}} \quad\text{(Oppenheim 2e)}\) ::: ::: {.column width=45%} Clinical tie-in (ECG)
• Example: $ $ input, $b=12$ ⇒ $= $
• Small $$ captures low-amplitude ST-segment changes (Webster MI 4e) ::: :::

Slide 6 — Quantization noise & SNR vs. bits

::: columns ::: {.column width=45%} Uniform noise model

• Noise power: \(\sigma\_q^2=\frac{\Delta^2}{12} \quad\text{(Oppenheim 2e)}\)
• Full-scale sinusoid SNR: \(\mathrm{SNR}\_{\mathrm{dB}}\approx 6.02\,b + 1.76 \quad\text{(Jayant & Noll, ISBN: 0-13-211913-7)}\) ::: ::: {.column width=45%} Clinical tie-in
• 12-bit ⇒ $+1.76 $
• EEG (tens of $$V): often benefits from $b$ plus low-noise analog front-end (Webster MI 4e) ::: :::

Slide 7 — Input range, gain, and clipping

• Set front-end gain so expected peaks approach full-scale without saturation
• Clipping distorts morphology and invalidates $SNR$ assumptions (Oppenheim 2e)
• Clinical tie-in (EMG): $ $ at skin ⇒ gain $$ to map to $ $ ADC; motion bursts can clip ⇒ add headroom (Webster MI 4e)
• Use limiter or wider range if artifacts are frequent

Slide 8 — Brief companding (telemetry)

::: columns ::: {.column width=45%} Idea

• Nonlinear mapping to emphasize low amplitudes before quantization/transmission
• $$-law: $F(x)=(x)$, $|x|$ (Jayant & Noll)
• Typical parameters: $$ (North America), $A=87.6$ (Europe) ::: ::: {.column width=45%} Clinical tie-in
• Low-bandwidth telemetry of voice/biomedical waveforms: preserves small ECG/PPG fluctuations at cost of large-signal fidelity
• Standards context: ITU-T G.711 $$-law/$A$-law PCM (identifier: G.711) ::: :::

Slide 9 — Plot: SNR vs. bit depth (formula-based)

• Deterministic illustration from $b=620$ using $6.02,b+1.76$
• Use when sizing ADC resolution given noise floor and clinical SNR target
• Assumption: full-scale sinusoid; real biometrics may deviate (Jayant & Noll)
• See code cell (matplotlib only)

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(7)  # determinism (even though not used)
b = np.arange(6, 21)
snr_db = 6.02*b + 1.76

plt.figure(figsize=(6,4))
plt.plot(b, snr_db, marker='o')
plt.xlabel("Bit depth b")
plt.ylabel("SNR (dB)")
plt.title("Quantization-limited SNR vs. bits")
plt.grid(True)
plt.tight_layout()

SNR(dB) versus ADC bit depth using SNR ≈ 6.02 b + 1.76 (Jayant & Noll, ISBN: 0-13-211913-7).

Slide 10 — References (IDs included)

• Oppenheim, A. V., Willsky, A. S., Nawab, S. H., Signals and Systems, 2nd ed., Prentice Hall, ISBN: 978-0138147570.
• Shannon, C. E., “Communication in the presence of noise,” Proc. IRE, 37(1), 1949, DOI: 10.1109/JRPROC.1949.232969.
• Nyquist, H., “Certain topics in telegraph transmission theory,” Trans. AIEE, 47, 1928, DOI: 10.1109/T-AIEE.1928.5055024.
• Jayant, N. S., Noll, P., Digital Coding of Waveforms, Prentice-Hall, 1984, ISBN: 0-13-211913-7.
• Webster, J. G. (ed.), Medical Instrumentation: Application and Design, 4th ed., Wiley, ISBN: 978-0471676003.
• ITU-T Recommendation G.711: Pulse Code Modulation (PCM) of voice frequencies (identifier: G.711).